Convergence and Optimal Buffer Sizing for Window Based AIMD Congestion Control

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Based AIMD Congestion Control

Konstantin Avrachenkov, Urtzi Ayesta, Alexei Piunovskiy

To cite this version:

Konstantin Avrachenkov, Urtzi Ayesta, Alexei Piunovskiy. Convergence and Optimal Buffer Sizing

for Window Based AIMD Congestion Control. [Research Report] RR-6142, INRIA. 2007, pp.39.

�inria-00136205v2�

inria-00136205, version 2 - 13 Mar 2007

a p p o r t

d e r e c h e r c h e

9

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2

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E

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Thème COM

Convergence and Optimal Buffer Sizing for Window

Based AIMD Congestion Control

Konstantin Avrachenkov — Urtzi Ayesta — Alexei Piunovskiy

N° 6142

Konstantin Avra henkov

∗

,Urtzi Ayesta

†

, AlexeiPiunovskiy

‡

ThèmeCOMSystèmes ommuni ants ProjetMaestro

Rapportdere her he n°6142February200739pages

Abstra t: Westudytheintera tionbetweentheAIMD(AdditiveIn reaseMultipli ative De rease) ongestion ontrol and a bottlene k router with Drop Tail buer. We onsider the problem in the framework of deterministi hybrid models. First, we show that the hybridmodeloftheintera tionbetweentheAIMD ongestion ontrolandbottlene krouter always onvergestoa y li behavior. We hara terizethe y les. Ne essaryandsu ient onditions for the absen e of multiple jumps of ongestion window in the same y le are obtained. Then, weproposean analyti alframework for the optimal hoi e of therouter buersize. Weformulate theproblem of theoptimal routerbuer size asamulti- riteria optimizationproblem,inwhi htheLagrangefun tion orrespondstoalinear ombinationof theaveragegoodputandtheaveragedelayin thequeue. Thesolutionto theoptimization problem provides further eviden e that the buer size should be redu ed in the presen e oftra aggregation. Ouranalyti alresultsare onrmedby simulationsperformedwith SimulinkandtheNSsimulator.

Key-words: TCP/IPmodeling,routerbuersizing,deterministi hybridmodelapproa h, multi- riteriaoptimization

TheworkwassupportedbyFran eTele omR&DGrantModélisationet GestionduTra Réseaux Internet no.42937433.

∗

INRIASophiaAntipolis,Fran e,e-mail:k.avra henkovsophia.inria.fr.

†

LAAS-CNRS,Fran e,e-mail:urtzilaas.fr.

‡

Résumé : Nous étudions l'intera tion entre le ontrle de ongestion AIMD (Additive In rease Multipli ative De rease) et le routeur ave le buer de type Drop Tail. Nous onsidérons eproblème dans le adre desmodèles hybridesdéterministes. D'abord, nous prouvons que le modèle hybride de l'intera tion entre la ntrole de ongestion AIMD et lerouteur de goulotd'étranglement onvergetoujours àun omportement y lique. Nous ara térisonsles y les. Des onditionsné essaires et susantes pour l'absen e des sauts multiplesdelafenêtrede ongestiondanslemême y lesontobtenues. Puis,nousproposons un adreanalytiquepourle hoixoptimaldelatailledubuerdurouteur.Nousformulonsle problèmedu hoixoptimaldelatailledubuerdurouteur ommeproblèmed'optimisation multi- ritère, dans lequel la fon tion de Lagrange orrespond à une ombinaison linéaire dutauxmoyen de transmissionet ledélai moyendans lebuer. La solutionauproblème d'optimisation fournit davantage d'éviden e que la taille du buer du routeur doit être réduiteen présen edel'agrégationdutra . Nosrésultatsanalytiques sont onrmés par dessimulationsee tuéesave Simulinket lesimulateurNS.

Mots- lés : lemodèledeTCP/IP, hoixoptimaldelatailledubuer,modèleshybrides déterministes,l'optimisationmulti- ritère

1 Introdu tion

Most tra in the Internet is governed by TCP/IP (Transmission Control Proto ol and InternetProto ol)[1,14℄. Datapa ketsofanInternet onne tiontravelfromasour enode toadestinationnodeviaaseriesofrouters. Somerouters,parti ularlyedgerouters, expe-rien eperiodsof ongestionwhenpa ketsspendanon-negligibletimewaiting intherouter buerstobetransmittedoverthenexthop. TCPproto oltriestoadjustthesendingrateof asour etomat htheavailablebandwidthalongthepath. Duringtheprin ipleCongestion Avoidan ephasethe urrentTCPNewReno versionusesAIMD(AdditiveIn rease Multi-pli ativeDe rease)binaryfeedba k ongestion ontrols heme. Intheabsen eof ongestion signalsfrom thenetwork TCPin reases ongestionwindowlinearlyin time, anduponthe re eption of a ongestion signal TCP redu es the ongestion window by a multipli ative fa tor. Congestionsignals anbeeitherpa ketlossesorECN(Expli itCongestion Noti a-tions)[21℄. AtthepresentstateoftheInternet, nearlyall ongestionsignalsaregenerated bypa ketlosses. Pa kets anbedroppedeitherwhentherouterbuerisfullorwhenAQM (A tiveQueueManagement)s hemeisemployed[10℄. Givenanambiguityin the hoi eof theAQMparameters[7, 16℄,sofarAQMisrarely usedin pra ti e. On theotherhand,in thebasi Drop Tail routers,the buersize isthe onlyone parameterto tuneapart ofthe router apa ity. In fa t, thebuer size is one of few parametersof the TCP/IP network that anbemanagedbynetwork operators. This makesthe hoi eof therouterbuersize averyimportantproblemintheTCP/IPnetwork design.

Thepaperis omposedoftwoprin ipleparts. Intherstpart(Se tions2-5)weanalyze theintera tionbetweentheAIMD ongestion ontrolandthebottlene krouterwithDrop Tail buer. This intera tion an be adequately des ribed by hybrid modeling approa h. Thereareseveralhybridmodelsoftheintera tionbetweenTCPandthebottlene krouter [4,6,13℄. Hereweanalyzethemodelof[13℄. Toouropinion,thismodeltakesintoa ount allessentialdetailsofTCPandatthesametimeleadstoatra tableanalysis. Weshowthat thesystemalways onvergestoalimitingbehavior. Inparti ular,wedemonstratethattwo dierentlimitingregimes an oexistandthe onvergen etooneortotheotherdependson theinitial onditions. Then,weprovidene essaryand su ient onditionsfortheabsen e of subsequent pa ket losses. The absen e of subsequent pa ket losses benets the TCP performan easwellasthequalityof servi eforendusers. Wenotethatin [13℄thereisno hara terizationoflimitingregimes. Furthermore,in [13℄onlyasu ient onditionforthe absen eofmultiplejumpswasobtainedandthesu ient onditionof[13℄islooseforsome valuesofthede reasefa tor.

Inthese ondpartofthepaper(Se tions6-7)westudytheoptimal hoi eof thebuer size in thebottlene k routers. There aresome empiri alrules forthe hoi e of therouter buersize. Therstproposed ruleofthumb forthe hoi e oftherouterbuersize wasto hoosethebuersizeequaltotheBDP(Bandwidth-DelayProdu t)oftheoutgoinglink[23℄. Thisre ommendationisbasedonveryapproximative onsiderationsandit anbejustied onlywhenarouterissaturatedwithasinglelong-livedTCP onne tion. Thenextapparent questiontoaskwashowoneshouldsetthebuersizeinthe aseofseveral ompetingTCP

the in rease of the buer size until a ertain threshold value. After that threshold value thefurtherin reaseofthebuersizedoesnotimprovethelinkutilizationbutin reasesthe queueing delay. Then, two ontradi toryguidelines for the hoi e of the buer size have beenproposed. In[17℄a onne tion-proportionalbuersizeallo ationisproposed,whereas in [3℄ it wassuggested that thebuer size should be set to the BDP of theoutgoing link dividedbythe squareroot of thenumberofTCP onne tions. A rationalefor theformer re ommendation is that in order to avoid a high lossrate the buer must a ommodate at leastfew pa kets from ea h onne tion. Andarationalefor thelatterre ommendation is basedonthe redu tionof thesyn hronizationof TCP onne tions when thenumberof onne tionsin reases. Then,[3,17℄werefollowedbytwoworks[8,11℄whi htrytore on ile these two ontradi toryapproa hes. Inparti ular, theauthors of[8℄ re ommendto follow theruleof[3℄forarelativelysmallnumberoflong-lived onne tionsand,whenthenumber of long-lived bottlene ked onne tions is large, to swit h to the onne tion-proportional allo ation. One of the main on lusions of [11℄ is that there are no lear riteria for the optimization of the buer size. Then, the author of [11℄ proposed a general avenue for resear h on the router buer sizing: Find the link buer size that a ommodates both TCPandUDPtra . WenotethatUDP(UserDatagramProto ol)[20℄doesnotuseany ongestion ontrolandreliableretransmissionanditismostlyemployedfordelaysensitive appli ationssu hasInternetTelephony. Werefertheinterestedreaderto[24℄andreferen es thereinformoreinformationontheproblemofoptimal hoi e ofbuersize.

All the above mentioned works on the router buer sizing are based on quite rough approximationsandstri tly speakingdonottakeintoa ountthefeedba knatureofTCP proto ol. Here we propose a mathemati ally solid framework to analyze the intera tion of TCP with the nite buer of an IP router. In parti ular, we state a riterion for the hoi e of theoptimal buer size in a mathemati al form. Ouroptimization riterion an be onsidered as a mathemati al formalization of the lingual riterion proposed in [11℄. Furthermore, thePareto set obtainedfor ourmodel allowsus to dimension the IP router buersizetoa ommodatebothdatatra andrealtimetra .

All proofsareprovidedin theAppendix.

2 Mathemati al model

The window based binary feedba k ongestion ontrol anbe des ribed by two fun tions

f (w)

and

G(w)

. Fun tion

f (w)

denes the in reaseprole of the ongestion window and fun tion

G(w)

represents the redu tion of the ongestion window upon the re eption of ongestion noti ation. Namely, in theabsen e of ongestionnoti ation the evolutionof ongestionwindow

w(t)

isdes ribedbythedierentialequation

dw

dt

=

f (w)

T + x(t)/µ

,

(1)

where

T

is the two way propagation delay,

x(t)

is the amount of data in the bottlene k

to theRound Trip Time(RTT) when theamountof theenqueued data in thebottlene k routerbueris

x(t)

attimemoment

t

. Thus,fun tion

f (w)

determinesthein reaseofthe ongestionwindowperoneRound Trip Time. Thesendingrate

λ(t)

ofthe window based ongestion ontrolisgivenby

λ(t) =

w(t)

T + x(t)/µ

.

(2)

Wewould liketo emphasizethat here thetime parameter

t

orrespondsto the lo al time observedat therouter.

WestudyaDropTailbuerwithsize

B

. If

x(t) < B

,the ongestionwindow

w

in reases a ordingto(1). When

x

rea hes

B

attime

t

∗

,i.e.

x(t

∗

_{) = B}

,thebuerstartstooverow. Theoverowofthebuerwillbenoti edbythesenderonlyafterthetimedelay

δ = T +B/µ

. Uponthere eptionofthe ongestionsignalattime

t

∗

_{+ δ}

,the ongestionwindowisredu ed a ordingto

w(t

∗

+ δ + 0) = G(w(t

∗

+ δ − 0)).

(3)

Asweshallseebelow,

w

(resp.,

λ

) anrepresenteithera ongestionwindow(resp.,sending rate) for a single TCP onne tion or a total window (resp., total rate) of several TCP onne tions.

Consider

n

long-livedAIMDTCP onne tionsthatshareabottlene krouter. Denoteby

w

i

(t)

theinstantaneous ongestionwindowof onne tion

i = 1, ..., n

attime

t ∈ [0, ∞)

. In the aseoftheAIMD ongestion ontrol,if

x < B

theevolutionofthe ongestionwindow

w

i

isgivenbydierentialequation(1) with

f (w) = m

i

= const

. Ifwerestri tourselvestothe symmetri ase

T

i

= T = const

and

m

i

= m

0

= const

, thesumof all ongestion windows

w(t) =

P

n

i=1

w

i

(t)

alsosatises dierentialequation (1)with

f (w) = m

, where

m = nm

0

. Namely,wehave

dw

dt

=

m

T + x(t)/µ

,

(4)

dx

dt

=



λ(t) − µ,

if

0 < x(t) < B,

or

x(t) = 0

and

λ(t) ≥ µ,

or

x(t) = B

and

λ(t) ≤ µ;

0

otherwise, (5) where

λ(t)

isgivenby(2). Andif

x(t

∗

_{) = B}

atsometimemoment

t

∗

,the ongestionwindow isde reasedmultipli ativelyaftertheinformationpropagationdelay

δ = T + B/µ

asfollows:

w(t

∗

+ δ + 0) = β

k

_w(t

∗

+ δ − 0),

(6)

and onsequently,

G(w) = β

k

_w

for the AIMD ase. Usually,

k = 1

, but sometimes it is ne essaryto send several ongestion signalsin orderto redu e thesending ratebelowthe transmission apa ityofthebottlene krouter.

Sin ewe onsiderthe aseofequalpropagationdelays,thesyn hronizationphenomenon takespla e[10℄, and onsequently,the totalsending rateis alsoredu edby thefa tor

β

k

. Forinstan e,inTCPNewRenoversiontheredu tionfa tor

β

isequalto onehalf.

Letusmakethe hangeoftimes alea ordingto

ds

△

=

dt

T + x(t)/µ

.

andthe hangeofvariables:

v

= w/m,

△

y

= x/m.

△

Thenew time

s

anbe viewedasa ounter forRound Trip Times. Now thedynami s of thesystembetweenthejumps isdes ribedbyequations

dv

ds

= 1,

(7)

dy

ds

=



v(t) − y(t) − q,

if

0 < y(t) < b,

or

y(t) = 0

and

v(t) ≥ q,

or

y(t) = b

and

v(t) ≤ q + b;

0

otherwise,

(8) where

q = µT /m

is the maximal numberof pa kets that an be t in the pipe, in other wordsBandwidth-DelayProdu t(BDP) in pa kets, and

b = B/m

is themaximal number ofpa ketsthat anbetin therouterbuer. Let

s

∗

bethemomentin thenewtimes ale when omponent

y

rea hesvalue

b

. Then,equation(6)istransformedto

v(s

∗

+ 1 + 0) = β

k

_v(s

∗

+ 1 − 0),

(9) where

k = min{i : β

i

_v(s

∗

+ 1 − 0) < b + q}

.

Remark1 Be auseof the delay in the information propagation, the ongestion windowis redu edafterthe delay

δ = T + B/µ

intheoriginaltimes ale,or, equivalently, after

1

time unitinthe newtimes ale

s

. Thevalueof

k

issu hthat,after sending

k

ongestionsignals, the amountofdata

x (

and

y )

startstode rease.

3 Convergen e of the system traje tories

Thedynami sisdenedbythreeparameters

β, q

,and

b

,andthesystemtraje toryremains intheregion

Ω = {0 ≤ y ≤ b, v > 0}

,provided theinitial onditionisthere.

Supposeatraje torystartsat

s = 0

frominitial ondition

y

0

= b

,

β(q + b) ≤ v

0

< b + q

, 1 and

s

∗

isthe rstmomentwhen

y(s

∗

_{) = b}

. Let

v

1

= v(s

∗

_{+ 1 + 0).}

Weintrodu emapping

ϕ

su hthat

v

1

△

= ϕ(v

0

)

. Considertheiterations

v

i+1

△

= ϕ(v

i

)

,

i = 0, 1, ...

. Theorem1 There exists

lim

i→∞

v

i

= V (v

0

)

with

V (v) =



V

1

,

if

v ∈ [β(q + b), d];

V

2

,

if

v ∈ (d, q + b),

(10) for some onstant

d

. Inparti ular, oneof the above intervals an beempty.

1

Initial onditions outside the region

[β(q + b), q + b)

are of no interest be ause, after the very rst

Denition1 Suppose the traje tory starting at

s = 0

from initial ondition

y

0

= b

,

v

0

<

b + q

rea hes the same point, for the rst time, at some time moment

S ≥ 1

. Then this nite traje tory is alled a y le. A y le with omponent

y

remaining zero for a positive time interval is alled lipped

(

see Figure 1

)

. If a y le tou hes the axis

y = 0

only at a singlepoint,we allsu h y le riti al

(

seeFigure2

)

.

Corollary1 (fromTheorem1) Any y lehas asingletimemoment,whena(multiple) jump o urs.

The number

k

of instant jumps of omponent

v

is alled a y le order. We allsu h y les

k

- y lesforbrevity. Ifoneof theintervalsin (10)is emptythen onlyasingle y le exists (Figure 1). Otherwise, two y les exist simultaneously (Figure 3); their orders are twosubsequentpositiveintegers. A ordingto Theorem1,whi h y leisrealizeddepends ontheinitial onditions.

4 Properties of y les

Inthis se tion, we hara terize theshapeof y les. Inother words, for given parameters

β

,

q

and

b

,wewould liketoknowifthelimit y lesofthe systemtraje toriesare lipped orun lippedand what orders the y les have. Forxed values of

β

and

q

, wedene the followingquantities:

N

△

= min



i ≥ 1 :

β

i

1 − β

i

< q



;

(11)

D

= ln(1 − β

△

N

) +

2β

N

1 − β

N

;

(12)

C

△

= − ln(1 − β

N

_{) − β}

N

_;

(13)

θ

k

isthesinglepositivesolutionto equation

ln

θ

1 − e

−θ

+

β

k

_θ

1 − β

k

= q −

β

k

1 − β

k

,

k = N, N + 1;

(14)

b

0,k

△

=

θ

k

1 − e

−θ

k

− ln

θ

k

1 − e

−θ

k

− 1.

(15) Then,wedenethesetofquantitieswhi hdonotdependon

q

:

τ

k

isthesinglepositivesolutiontoequation

τ

1 +

β

k−1

_1−β

−β

k

k

(τ + 1)

= 1 − e

−τ

_,

_{k = 2, 3, . . .}

(16)

A

∗

k

△

=

β

k−1

_(τ

k

+ 1)

1 − β

k

;

(17)

q

∗

k

△

=

β

k

1 − β

k

(τ

k

+ 1) + ln

τ

k

1 − e

−τ

k

;

(18) Itis onvenienttoput

τ

1

, A

∗

1

and

q

∗

1

equalto

+∞

. Finally, in ase

q ≤ D

onehastosolve equation

e

−r

_{+ r − 1 = β}

N

_{(q + r + 1) − q.}

(19) Ithasnomorethantwopositivesolutions

r ≤ ¯r

whi hdene

b

△

= e

−r

_{+ r − 1; ¯b}

△

_{= e}

−¯

r

_{+ ¯}

_{r − 1,}

(20) Notethat

b ≤ ¯b

. If

q ≤ q

∗

N+1

then

q ≤ D

and

¯b ≥ A

∗

N+1

− q

.

We notethat all theabovedened quantities donotdepend on

b

. Thus, from nowon weassumethat

β

and

q

arexedandwearegoingtodes ribewhatkindof y lesexistfor dierentvaluesof

b

. Inotherwords,westudywhat ee ttherouterbuersize hasonthe limitingbehaviorofTCP/IP.Therearethree ases:

Case

A

∗

N+1

< q

. If

b ∈

h

0,

_1−β

β

N −1

N −1

− q

i

then only the y le of order

N

exists. In ase

N = 1

, we put

β

0

1−β

0

= +∞

forgenerality. Suppose

N > 1

. Thenfor

b ∈



β

N −1

1−β

N −1

− q, A

∗

N

− q

i

two y les,oforders

N

and

N − 1

existsimultaneously. For

b ∈



A

∗

N

− q,

β

N −2

1−β

N −2

− q

i

,thereexistsonlyasingle y leoforder

N − 1

. Andsoon;for

b > A

∗

2

− q

,only1- y leexists(seeFigure13).

The

N

- y leis lippedfor

b ∈ [0, b

0,N

)

. Cy lesoflowerordersareun lippedforallvalues of

b

,iftheyexist.

The

N

- y le tou hes the

v

-axisat a singlepointi

b = b

0,N

. Thus, if

b = b

0,N

there existsa riti al

N

- y le. No riti al y lesoflowerordersexist.

Example1 Letusillustrate thiswithanumeri alexample. Ifwetake

q = 0.9

and

β = 1/2

then

N = 2

,

A

∗

2

= 1.4965

,

A

∗

3

= 0.3910

. If

b ∈ [0, 0.1]

we have only 2- y les; if

b ∈

(0.1, 0.5965]

we have 1- y les and 2- y les

(

see Figure 3

)

; and if

b > 0.5965

we have only 1- y les. Forea h

b < b

0,2

= 0.0617

,there existsonly a lipped2- y le

(

seeFigure 1

)

. As one an see on Figure 2 , when

b = b

0,2

= 0.0617

,the 2- y le be omes riti al. All gures for thisexample havebeen plottedwithMATLABSimulink.

Case

q ≤ q

∗

N

+1

.

If

b ∈ [0, b)

,thenonlythe

N

- y leexists. If

b ∈ [b, A

∗

N+1

− q]

,then two y lesoforders

N

and

N + 1

existsimultaneously. For

b ∈ (A

∗

N+1

− q,

β

N −1

1−β

N −1

− q]

,again,onlythe

N

- y le exists.

The

N

- y leis lippedfor

b ∈ [0, b

0,N

)

; the

(N + 1)

- y leis lipped for

b ∈ [b, b

0,N +1

)

. These y lesbe ome riti alat

b = b

0,N

and

b = b

0,N +1

,respe tively. Cy lesoflowerorders

0

0.5

1

1.5

2

2.5

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

rescaled window, v

rescaled buffer occupancy, y

Instantaneous

window reduction

Figure1: Clipped2- y le. Case

A

∗

2

< q

.

0

0.5

1

1.5

2

2.5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

rescaled window, v

rescaled buffer occupancy, y

Instantaneous

window reduction

Figure2: Criti al2- y le. Case

A

∗

2

< q

.

0

1

2

3

4

5

6

7

8

0

0.5

1

1.5

2

2.5

3

rescaled time, s

rescaled window, v

Convergence to 2−cycle

Convergence to 1−cycle

0

1

2

3

4

5

6

7

8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

rescaled time, s

rescscaled buffer occupancy, y

Convergence to 2−cycle

Convergence to 1−cycle

0

0.5

1

1.5

2

2.5

3

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

rescaled window, v

rescaled buffer occupancy, y

Trajectory

with 2−cycle

Trajectory

with 1−cycle

Instantaneous

window reduction 1−cycle

Instantaneous

window reduction 2−cycle

Figure3: Co-existen eof1- y leand2- y le. Case

A

∗

2

< q

.

areun lippedforallvaluesof

b

,iftheyexist.If

N > 1

then,similarlytothe ase

A

∗

N

+1

< q

, theorderofthe y lede reasesas

b

in reasesabove

β

N −1

1−β

N −1

− q

(seeFigure13). Case

q

∗

N

+1

< q ≤ A

∗

N+1

. If

C ≤ A

∗

N

+1

− q

2

and

q ≤ D

, then everythingissimilar to the ase

q ≤ q

∗

N

+1

. The dieren eisthatthe

(N + 1)

- y leis lippedand annotbe riti al;itexistssimultaneously withthe

N

- y lefor

b ∈ [b, ¯b]

. If

b ∈

¯b,

β

N −1

1−β

N −1

− q

i

, only the

N

- y leexists. Thelatter intervalisnon-empty.

If

C > A

∗

N

+1

− q

or

D < q

,theneverythingisexa tlyasin ase

A

∗

N+1

< q

. 2

A tually

C

annotbeequalto

A

∗

N +1

− q

.

5 Conditions for the absen e of multiple jumps

The regime with multiple jumps is not desirable. The multiple jump orresponds to the lost ofmorethan onepa ketin asingle ongestion window. Subsequent pa ket losses an for eTCPtoswit hfromtheCongestionAvoidan eTCPphasetotheSlowStartphaseand leadtolengthytimeouts. Furthermore,theabsen eofsubsequentpa ketlossesisbene ial notonly for theTCP performan ebut also forthe qualityof servi e provided to the end users. Inthenexttheoremweprovidene essaryandsu ient onditionsfortheabsen eof multiplejumps,namely,we hara terizeallpossible aseswhenonlyasingle y leoforder

1

exists.

Theorem2 Thefollowingmutuallyex lusive onditionsfully hara teriseallpossible ases whenonlyasingle y leof order

1

exists:

(a)

β

1−β

≥ q

and

b + q > A

∗

2

; (b)

A

∗

2

< q

(

b

anbearbitrary); ( )

β

1−β

< q ≤ q

∗

2

and

b /

∈ [b, A

∗

2

− q]

; (d)

max

n

β

1−β

, q

∗

2

o

< q ≤ A

∗

2

− C

,

q ≤ D

and

b /

∈ [b, ¯b]

; (e)

max

n

β

1−β

, q

∗

2

o

< q ≤ A

∗

2

− C

,

q > D

(

b

anbearbitrary); (f)

max

n

β

1−β

, q

∗

2

, A

∗

2

− C

o

< q ≤ A

∗

2

(

b

an bearbitrary).

Inthefollowing orollaryweprovideasimplesu ient onditionforabsen eofmultiple jumps.

Corollary2 Condition

b + q > A

∗

2

is su ient for the absen e of y les of orders

k > 1

. (SeeFigure 13andCorollary 6.)

Re allthat

A

∗

2

dependsonlyon

β

. Inparti ular,if

β = 1/2

,

A

∗

2

= 1.4965

.

We would like to note that theabove su ient onditionis tighter than thesu ient ondition for the absen e of multiple jumps provided in [13℄:

b + q > 2β/(1 − β)

. To omparethese two onditions,weplot

A

∗

2

(β)

and

2β/(1 − β)

inFigure4andthedieren e

2β/(1 − β) − A

∗

2

(β)

inFigure5. Stri tlyspeakingwehave Proposition 1 Thedieren e

δ

△

=

_1−β

2β

− A

∗

2

isalwayspositive and

lim

β→1

δ = +∞

. Nevertheless,thesimplesu ient onditionof[13℄appearsto bequitegoodex eptfor valuesof

β

that aretoo losetoone.

0.5

0.6

0.7

0.8

0.9

1

0

20

40

60

80

100

120

140

160

180

200

β

A

₂

*

2

β

/(1−

β

)

Figure4: Comparisonof su ient onditions fortheabsen eofmultiple jumps.

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15

20

25

β

δ

=2

β

/(1−

β

)−A

2

*

Figure5: Thevalueof

2β/(1 − β) − A

∗

2

(β)

.

6 Pareto set for optimal buer sizing

Let us study what ee t has the hoi e of the buer size on the performan e of TCP. In parti ular,weareinterestedinoptimalbuersizing. Towardsthisgoal,letusformulatethe performan e riteria. Ononehand,weareinterestedtoobtainaslargegoodputaspossible. Thatis,weareinterestedtomaximizetheaveragegoodput

¯

g = lim

t→∞

1

t

Z

t

0

g(s)ds,

wheretheinstantaneousgoodput

g(t)

isdenedby

g(t) =



λ(t),

if

x(t) < B,

µ,

if

x(t) = B.

On theother hand,weare interested to makethe delay of data in thebuer as small as possible. That is, we are also interested to minimize the average amount of data in the buer

¯

x = lim

t→∞

1

t

Z

t

0

x(s)ds.

Clearly,thesetwogoalsare ontradi tory. Infa t,herewefa eatypi alexampleof multi- riteria optimization. A standard approa h to it is to onsider the optimization of one riterionunder onstraintsfor the other riteria (seee.g., [19℄). Namely, wewould liketo maximizethegoodputgiventhattheaverageamountofdatain thebuerdoesnotex eed a ertainvalue

Orwewould liketominimizethe average delaygiventhat theaveragegoodputis notless thana ertainvalue

f min{¯x : ¯g ≥ ¯g

∗

}.

(22) The solution to the above onstrained optimization problems an be obtained from the Pareto set. As is known, see e.g. [19℄, the Pareto set an be onstru ted by solvingthe optimizationproblem

max



lim

t→∞

1

t

Z

t

0

c

1

g(s) − c

2

x(s)ds



.

(23)

Tobemorepre ise,theParetoSetisformedbythepairsofobje tives

(¯

g, ¯

x)

thatsolve(23) fordierent

(c

1

, c

2

) ∈ R

2

+

. AnexampleofParetosetisgiveninFigure6. Ea hpointofthe Paretoset orrespondstoasolutionofoptimizationproblem(23)forsome hoi eof

c

1

and

c

2

. On eweobtainthePareto set,itisveryeasytodedu e solutionof problems(21)and (22). Forinstan e,ifonewantsthattheutilizationofthebottlene krouterwillbenotless than,say,95%,onehastobereadytoa eptthedelaysthat areequalorgreaterthan

x

∗

.

0

x

x

µ

µ

0.95

g

*

Figure6: Paretoset.

0

v

B

y

b

q

A

D

C

E

Figure7: Phases ofthe lipped y le. All three optimization problems (21), (22) and (23) an be regarded asmathemati al formulationof thelingual riterionndthelinkbuer sizethat a ommodatesbothTCP andUDPtra  givenin[11℄. Sin e UDPtra doesnot ontributemu hin termsofthe load,forthedesignofIP routersone anuseforinstan eoptimizationproblem(21)where thedelay onstraintisimposedbytheUDPtra .

Wenote thatherewedealwiththeoptimalimpulse ontrolproblemof adeterministi systemwithlong-run average optimality riterion. Tothebest of ourknowledgethere are no available results on su h type of problems in the literature. In prin iple, the ontrol poli y in ourmodel andependonthe urrentvaluesof

x

and

λ

. Inpra ti e,however,all urrentlyimplementedbuermanagements hemes(e.g., AQM,DropTail)send ongestion signalsbasedonlyonthestateofthebuer. Thus,wealsolimitourselvestothe asewhen the ontroldepends onlyontheamountofdatain thebuer. Furthermore,werestri tthe

ontrola tion onlytothe hoi eofthe buersize. Thus,the ontrolsignalisonlysentat themomentwhenthebuergetsfull.

The following theorem provides expressions forthe averagesending rate, goodputand queuesize under ondition

q > A

∗

2

,whi hguaranteestheabsen eofmultiplejumpsforany valueofthebuersize. Rememberthat

A

∗

2

dependsonlyon

β

(see(16),(17)). Inparti ular, theexpressionsallowusto plottheParetosetparameterizedbythebuersize.

Theorem3 Letthe ondition

µT /m > A

∗

2

besatised. Then,for

B ∈ [0, mb

0,1

]

theaverage sendingrate,goodput andbuer o upan yaregiven by

¯

λ =

m(1 − β

2

₎

2T

cycle



1 +

µT

m

+ S

CD



2

,

¯

g =

m

T

cycle

"

1

2

 µT

m

+ S

CD



2

−

β

2

2



1 +

µT

m

+ S

CD



2

+

µT + B

m

#

,

¯

x =

1

T

cycle

"

mT

Z

S

AB

0

y

AB

(s)ds +

Z

S

CD

0

y

CD

(u)du

!

+

m

2

µ

Z

S

AB

0

y

2

AB

(s)ds +

Z

S

CD

0

y

2

CD

(u)du +

B(µT + B)

m

2

!#

,

respe tively,where

T

cycle

isthe y leduration given by

T

cycle

= (1 − β)(1 +

µT

m

+ S

CD

)T +

B

µ

+

m

µ

Z

S

AB

0

y

AB

(s)ds +

Z

S

CD

0

y

CD

(u)du

!

,

with

y

CD

(u) = e

−u

+ (u − 1),

y

AB

(s) = [

B

m

+ (1 − β)(1 +

µT

m

) − βS

CD

]e

−s

+ (s − 1) + β(S

CD

+ 1) − (1 − β)

µT

m

,

where

S

CD

and

S

AB

arethe solutionsof the equations

e

−S

CD

_{+ S}

CD

− 1 =

B

m

,

 B

m

− βS

CD

+ (1 − β)(1 +

µT

m

)



e

−S

AB

_{+ S}

AB

+ βS

CD

− (1 − β)(1 +

µT

m

) = 0.

For

B ∈ (mb

0,1

, ∞)

,wehave

¯

λ =

m

2T

cycle

1 + β

1 − β

(s

1

+ 1)

2

_,

¯

g = µ,

¯

x =

1

T

cycle



mT

Z

s

1

0

y(s)ds +

m

2

µ

Z

s

1

0

y

2

(s)ds +

B(µT + B)

m

2



,

where

T

cycle

= T (s

1

+ 1) +

m

µ

Z

s

1

0

y(s)ds +

B

m



with

y(s) =



1 +

µT + B

m

− v

0



e

−s

+ (s − 1) + v

0

−

µT

m

,

where

v

0

and

s

1

are denedby

(

28

)

and

(

29

)

with

k = 1

.

Example2 LetusillustratetheParetosetforaben hmarkexampleoftheTCP/IPnetwork reatedwiththehelpofNS-2simulator

[

18

]

. Thenetwork onsistsofasinglebottlene klink of apa ity

µ = 10M bps

whi h issharedby

n

long-livedTCP onne tions. The propagation delay for ea h onne tion is

T = 0.24s

and

β = 1/2

. The pa ket size is

4000bits

. Thus, we have that

m

0

= 4000bits

as well. In Figure 8 we plot the Pareto set for

n = 10

(and

m = nm

0

= 40, 000

)usingthe formulaeof Theorem 3andmeasurementsobtainedfromNS simulations. Asone ansee, two urves mat hwell. In Figure 9,again usingthe formulae of Theorem 3,weplot the averagegoodput andthe average sending rate asfun tionsof the buersizefor

n = 60

.

Wenotethat

g ≤ µ

¯

always,buttheaveragesendingrate

λ

¯

anex eedtherouter apa ity

µ

(seeFigure9). Nevertheless,asthenextProposition 2states,thedieren ebetweenthe averagesendingrateandtherouter apa itygoestozeroas

B

in reases. Inparti ular,this means that when the Drop Tail router is used, the rate of lost (and then retransmitted) informationeventuallydiminishesto zeroasthebuersizein reases.

Proposition 2 When

B → ∞

,the dieren e

∆ = ¯

λ − µ

approa hes zerofrom above.

7 Minimal buer size for the full system utilization Inthe aseofmultipleTCP onne tions ompetingforresour eofthebottlene krouterwe have

m = nm

0

. Here

n

isthenumberof ompetingTCP onne tions. Letusstudyhowthe minimal buersize forthefull systemutilization,

B

0,N

, depends on

n

or,equivalently,on

m

.

B

0,N

is thebuersize orrespondingto s enariowhentheParetoset tou hesthelevel

µ

(seeFigure6). It orrespondsalsotothe riti al y leofminimalorder.

Proposition 3

(a)

Foraxed

N

,the value of

B

0,N

= mb

0,N

de reasesas

m

in reases.

0

50

100

150

200

250

300

0

1

2

3

4

5

6

7

8

9

10

11

x 10

6

average occupancy (packets)

average goodput (bits/sec)

NS−2 simulations

Theoretic curve

Figure 8: Pareto set: Numeri al ulations andNS-2simulations.

0

500

1000

1500

2000

2500

8

8.5

9

9.5

10

10.5

x 10

6

buffer length, B (packets)

bits/sec

average sending rate

average goodput

Figure 9: Non-monotoni ity of the average sendingrate.

Corollary3 The buersize

B

0,N

ofthe minimal order riti al y leisapie e-wise dier-entiablefun tion of

m

,de reasing onthe intervals

[m

i

, m

i+1

)

;

lim

m→m

i+1

−0

B

0,N

(m) < B

0,N

(m

i+1

),

i = 0, 1, 2, . . .

Here

m

i

△

= µT (1 − β

i

_)/β

i

;the valueof

N

equals

i + 1

onthe interval

[m

i

, m

i+1

) (

see

(

11

) )

. Moreover,

lim

m→m

N

−

0

B

0,N

= 0

,

lim

m→m

N

−

0

dB

0,N

dm

= 0

,

lim

m→0+

B

0,1

= µT (1 − β)/β

,

lim

N →∞

m

N −1

B

0,N

(m

N −1

) = 0.5(µT (1 − β))

2

andhen e

lim

m→∞

B

0,N

= 0

.

Example2( ntd.) In Figure 10weplot the buersize

B

0,N

ofthe minimal order riti al y leandthe urve

f (m) = (1 − β)

2

_{(µT )}

2

_/(2m)

for

µT = 2.4 × 10

6

_{bits (600packets)}

. The urve

f (m)

indeedapproa hes fast the lo al maximaof

B

0,N

as

m

in reases. InFigure 11 wemake azoomon the interval withsmaller values of

m

. As one ansee, when

m

goes to zero, the valueof

B

0,N

approa hes 600pa kets,whi h isthe BDPin thisnetwork example.

We note that by Corollary3for small values of

m

theminimal buer size for the full system utilization is approximatelyequal to

µT

, BDP of the bottlene k link. This is in agreementwiththeempiri al on lusionof[23℄. In[3℄theauthorssuggestedthatthe mini-malbuersize forthefullsystemutilizationshouldde reaseas

(µT )/

√

_n

asthenumberof onne tions

n

in reases. Wenotethat theauthorsof[3℄haveassumedthat the ompeting TCP onne tions arenot syn hronized. That is, only asingle onne tionredu es its on-gestionwindowwhenthebuerbe omesfull. Inourmodelweassumefullsyn hronization of ompeting TCP onne tions. Namely, whenthebuerisfull, all onne tions simultane-ouslyredu etheir ongestion windows. Weexpe tthat thesituationin realnetworksis in betweenthesetwoextremes. And thus, themodel of[3℄providesan upperbound and our modelprovidesalowerbound. Furthermore,itwasbelievedpreviouslythatifthe

ompet-thefullsystemutilization. FromFigure11one anseethattheminimalbuerrequirement de reaseswithin reasing

m

(or,equivalently,within reasing

n

)eveninthe aseof omplete syn hronization. Finally,wewouldliketomentionthatthevalueof

B

0,N

isnon-monotonous with respe t to

m

, eventhough it eventually de reasesto zero(see Figure10). Curiously enough,theexperimentsof[24℄withtherouter,runningFreeBSDdummynetsoftware,have alsoshownthenon-monotonous behaviorof theminimal buerrequirementin the aseof syn hronized onne tions(seeFigure1in[24℄).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 10

4

0

10

20

30

40

50

60

70

80

90

100

n

packets

B

0N

(

µ

T)

2

(1−

β

)

2

/(2m

0

n)

3−cycle

4−cycle

5−cycle

Figure10: Theminimalbuersizeforthefull systemutilization.

0

10

20

30

40

50

60

0

100

200

300

400

500

600

n

packets

\mu*T/(m_0\sqrt(n))

B

0N

Figure11: Theminimalbuersizeforthefull systemutilization (zoom).

8 Con lusions

In this paper we have studied the intera tion between AIMD Congestion Control and a bottlene krouterwithDropTailbuer. Wehaveusedthehybridmodelingapproa h. Itis demonstratedthatthesystemalways onvergestoa y li behavior. Thelimit y leshave beenfully hara terized. Inparti ular,wehaveobtainedne essaryand su ient ondition for the absen e of y les with multiple jumps and a simple but tightsu ient ondition. Then,wehaveformulatedtheproblemof hoosingthebuersizeofroutersintheInternetas amulti- riteriaoptimizationproblem. Inagreementwithpreviousworks,ourmodelsuggests that asthenumberof long-livedTCP onne tions sharingthe ommonlink in reases,the minimalbuersize requiredto a hievefulllink utilization de reases. However,inthe ase of syn hronized onne tions, thede rease is not monotonous and slowerthan theinverse ofthesquareroot ofthenumberof onne tions. ThePareto setobtainedwiththehelp of ourmodelallowsustoevaluatetheIProuterbuersizeinordertoa ommodaterealtime tra aswellasdatatra . Thesimulations arriedoutwiththehelpofSimulinkandNS

framework toother ongestion ontrolme hanisms,su hasMIMD, HighSpeedTCP,TCP Westwoodappearstobeafruitfuldire tion forfuture resear h.

Appendix Un lipped y les.

Inthis andthenextsubse tion,weignoretherequirementthat

y ≥ 0

. Thus dynami s isdes ribedbyequations















dv

ds

= 1;

dy

ds

=







v − y − q,

if

y < b,

or

y = b

and

v ≤ A;

0

otherwise, (24) where

A

= b + q.

△

Thejumps o ura ordingto(7)asbefore.

Denition2 Let

y

0

= b

and

v

0

< A

bethe initial onditions. A pie e of traje tory on the timeinterval

[0, s

∗

+ 1 + 0]

is alledapseudo- y leoforder

k

(see(7)). If

v(s

∗

+ 1 + 0) = v

0

thenthe pseudo- y leis alleda

k

- y le.

Later,itwillbeshownthatifa lipped

k

- y leexiststhentheun lipped

k

- y leexists, too(Corollary6). Clearly,(24)hasasinglesolution



v(s) = v

0

+ s;

y(s) = (1 + q + y

0

− v

0

)e

−s

+ s − 1 + v

0

− q.

(25) Theorem4 An(un lipped)

k

- y leexistsi

A ∈



_β

k

1 − β

k

, A

∗

k



,

(26) where

A

∗

k

△

=

(

β

k−1

_(τ

k

+1)

1−β

k

,

if

k > 1,

∞,

if

k = 1

(27) and,for

k > 1

,

τ

k

isthe singlepositive solutionto(16 ).

Proof. Obviously, parameters of a

k

- y le,

v

0

and time interval

s

1

an be found from equations

whi h areequivalentto

v

0

=

β

k

_(s

1

+ 1)

1 − β

k

.

(28)

1 − e

−s

1

₌

s

1

1 + A −

β

k

(s

1

+1)

1−β

k

.

(29)

A

k

- y le exists i (29) has a positive solution and

v

0

given by (28) satises inequality

v

0

≥ βA

. (Otherwise,if

v

0

< βA

, thereis noneed to redu e

v

somanytimes.) Equation (29)hasapositivesolutioni

1 + A −

β

k

1 − β

k

> 0

and

d

ds





s

1 + A −

β

k

1−β

(s+1)

k

















s=0

< 1

-s

6

0

1

1 − e

−s

s

1+A−

βk(s+1)

1−βk

Figure12: Graphi alsolutiontoequation(29).

(seeFig.12),or,equivalently,i

β

k

(1 + A) < A.

(30) Put

K

△

= min{i ≥ 1 : β

i

<

A

Lemma1 If

v

0

∈ [βA, A)

then, starting from

v

0

,

y

0

= b

,the next instant seriesof

K + 1

jumpsresults in the value

v < A

. Hen ethe order of any y le annotex eed

K + 1

(and learly annotbesmaller than

K

).

Proof. Suppose

ˆ

v

0

= βA

. Then,after thenextinstantseries of

K + 1

jumps,thevalue

ˆ

v

isnotsmallerthan

v

. Toputitdierently,

v ≤ β

K+1

[βA + ˆ

s + 1],

(32) where

s

ˆ

solvesequation

(1 + A − βA)e

−s

+ s − 1 + βA = A

⇐⇒

s

1 + (1 − β)A

− 1 + e

−s

_{= 0.}

(33) Ifwesubstitute

˜

s

△

=

A + 1

β

− βA − 1 <

A

β

K+1

− βA − 1

into(33)weobtain,usingequality

A =

β˜

s+β−1

1−β

2

:

˜

s

1 + (1 − β)A

− 1 + e

−

˜

s

₌

s(1 + β)

˜

β(2 + ˜

s)

− 1 + e

−

˜

s

_>

2˜

s

1 + ˜

s

− 1 + e

−

˜

s

_{> 0.}

When

s

in reases from zero, the lefthand side of (33) initially de reases from zero and in reasesthereafter. Hen e

s > ˆ

˜

s

and(32)implies

v < β

K+1

_{[βA + ˜}

_{s + 1] < β}

K+1

_{[βA +}

A

β

K+1

− βA − 1 + 1] = A.

Lemma2 Suppose

β ∈ (0, 1)

isxedand onsider fun tion

f

k

(A)

△

= (1 − β

k

_{)A − β}

k−1

_(s

1

+ 1),

(34)

where

s

1

solves (29). Thedomain of

f

isgiven by(30 ). Then (a)

df

k

(A)

dA

> 0

;

(b)

f

1

(A) < 0

for all

A >

β

1−β

;

( )

∀k > 1

equation

f

k

(A) = 0

hasasinglenitesolution

A

∗

k

givenby(27 );

A

∗

k

de reases as

k

in reases. (d)

∀k > 1 A

∗

k

>

β

k−1

1−β

k−1

;

∀k > 2

,

A

∗

k

≤

β

k−2

1−β

k−2

.

Proof. (a)A ordingtotheruleof impli itdierentiation,appliedtoequation



1 + A −

β

k

_(s

1

+ 1)

1 − β

k



1 − e

−s

1

_{− s}

1

= 0,

wehave

ds

1

dA

= −

(1 − e

−s

1

₎

2

_{(1 − β}

k

₎

e

−s

1

_s

1

(1 − β

k

) − (1 − e

−s

1

)

2

β

k

− (1 − β

k

)(1 − e

−s

1

)

.

Thedenominator equals

−(1 − e

−s

1

_{− s}

1

e

−s

1

) − β

k

(s

1

e

−s

1

− e

−s

1

+ e

−2s

1

)

< β

k

(e

−s

1

_{− e}

−

2s

1

_{− s}

1

e

−

2s

1

) − (1 − e

−s

1

− s

1

e

−s

1

)

= (1 − e

−s

1

_{− s}

1

e

−s

1

)(βe

−s

1

− 1) < 0;

hen e

ds

1

dA

> 0

for

s

1

> 0

. Now

df

k

(A)

dA

= (1 − β

k

_{) − β}

k−1

ds

1

dA

=

(1 − β

k

_{)[(1 − β}

k

_)(s

1

e

−s

1

− 1 + e

−s

1

) + β

k−1

(1 − e

−s

1

)

2

(1 − β)]

s

1

e

−s

1

(1 − β

k

) + (1 − e

−s

1

)(β

k

e

−s

1

− 1)

.

Thedenominator isnegative(seeabove). Thenominatordoesnotex eed

(1 − β

k

_{)(1 − β)[(s}

1

e

−s

1

− 1 + e

−s

1

)(1 + β

k−1

) + β

k−1

(1 − e

−s

1

)

2

]

= (1 − β

k

_{)(1 − β)[(s}

1

e

−s

1

− 1 + e

−s

1

) + β

k−1

e

−s

1

(s

1

− 1 + e

−s

1

)].

Thebothtermsin thelattersquarebra ketarenegativefor

s

1

> 0

. Hen e

df

k

(A)

dA

> 0

. (b)Itissu ienttoprovethat

s

1

> S

△

= A(1 − β) − 1,

where

s

1

solves(29)at

k = 1

.

Case

S < 0

istrivial,thusassumethat

S > 0

. Letussubstitute

S

intothebothsidesof (29)andestimatethedieren e:

S

1 + A −

β(S+1)

1−β

− 1 + e

−S

=

S

1 +

S+1

1−β

−

β(S+1)

1−β

− 1 + e

−S

= e

−S

−

_{S + 2}

2

< 0,

be ausefun tion

(S + 2)e

−S

de reasesfrom2at

S = 0

. To ompletethispartoftheproof, itissu ienttonoti ethat,ontheinterval

therighthandsideof(29)issmallerthanthelefthandsidei

S < s

1

. ( ) The rst partis obvious:

A

∗

k

is given by(27), provided equation (16)has a single positivesolution. Thelatterstatementfolowsfromthefa t thatfun tion

g(τ ) = (1 − e

−τ

_{)(1 + α(τ + 1))/τ}

de reasesto

lim

τ →∞

g(τ ) = α

,startingfrom

lim

τ →0

g(τ ) = 1 + α

. Here

α

=

△

β

k−1

_{− β}

k

1 − β

k

.

(35) Indeed,

dg

dτ

=

e

−τ

_{[1 + α + τ (1 + α + ατ )] − (1 + α)}

τ

2

< 0

in ase

α < 1

,and

α =

β

k−1

1 + β + . . . + β

k−1

≤

1

1 + 1/β

< 1/2.

(36) Now, look what happens as

k

in reases. Obviously, fun tions

β

k−1

1−β

k

=

β

k

1−β

k

·

1

β

and

α =

_1−β

β

k

k

(

1

β

− 1)

(see(35))de rease. A ordingto(27)itremainstoprovethat

τ

k

givenby (16)in reaseswith

α

. Werewrite(16)as

(1 + α(τ + 1))(1 − e

−τ

_{) − τ = 0.}

Hen e

dτ

k

dα

= −

(τ

k

+ 1)(1 − e

−τ

k

)

α(1 − e

−τ

k

) + (1 + α(τ

k

+ 1))e

−τ

k

− 1

= −

(τ

k

+ 1)

2

(1 − e

−τ

k

)

2

h(τ

k

)

,

where

h(τ ) = −2 + 3e

−τ

_{− e}

−

2τ

_{+ τ e}

−τ

_{+ τ}

2

_e

−τ

_.

(Wehavesubstituted

α =

τ

k

−1+e

−τk

(1−e

−τk

)(τ

k

+1)

.) Weintendtoprovethat

dh

dτ

= −2e

−τ

_{+ 2e}

−

2τ

_{+ τ e}

−τ

− τ

2

e

−τ

< 0

(37)

when

τ > 0

. Clearly(37)holdsfor

τ ≥ 1

. Suppose

τ ∈ (0, 1)

. Then

d

2

_h

dτ

2

= 3e

−τ

_{− 4e}

−2τ

− 3τe

−τ

_{+ τ}

2

_e

−τ

_{< 3e}

−τ

_{− 4e}

−2τ

− 2τe

−τ

_{= e}

−τ

_{(3 − 4e}

−τ

_{− 2τ).}

Expressionin thebra kets hasa negative maximumat

τ = ln 2

. Therefore,

d

2

h

dτ

2

< 0

and

dh

dτ

< 0

. Finally,

h(τ ) < 0

forall

τ > 0

,be ause

h(0) = 0

. (d) To estimate

A

∗

k

from below, weuse statement(a): it is su ientto establish that

f

k



β

k−1

1−β

k−1



< 0

,ie

s

1

+ 1 >

1−β

k

1−β

k−1

⇐⇒

S

1+

βk−1

1−βk−1

−

βk(S+1)

1−βk

< 1 − e

−S

for

S =

1−β

k

1−β

k−1

− 1

. (Theargumentissimilarto (b).) But

S

be ausefun tion

e

−S

_{(1 + S)}

de reasesfrom

1

at

S = 0

. Finally, in ase

k > 2

, suppose

A

∗

k

>

β

k−2

1−β

k−2

. Then forparameters values

β

and

A ∈



β

k−2

1−β

k−2

, A

∗

k



wehavethat(30)holdsfor

k − 2

,

k − 1

,and

k

andsimultaneously

f

k

(A) < 0

,

f

k−1

(A) < 0

,

f

k−2

(A) < 0

: see (a) and ( ). A ording to the beginning of the proof of Theorem1, y lesoforders

k

,

k − 1

,and

k − 2

existwhi h ontradi tsLemma1.

Nowwe aneasily nish the proofofTheorem4. Supposea

k

- y le exists. Then, a - ordingto(30),

A >

β

k

1−β

k

. Lemma2guaranteesthat

A

∗

k

>

β

k−1

1 − β

k−1

>

β

k

1 − β

k

,

and,aswasmentionedearlier,inequality

v

0

≥ βA

mustbevalid(see(28)),whi his equiv-alent to

A ≤ A

∗

k

. Finally, if (26) holds then (29) has apositive solution (see (30) ) and

v

0

≥ βA

;hen ea

k

- y leexists.

-6

Case

A

∗

N +1

≥ q

:

@

R

₆

Case

A

∗

N +1

< q

:

0

5- y leexists

r

A

∗

5

− q

r

4- y le exists

A

∗

4

− q

3- y leexists



r

A

∗

3

− q

β

3

1−β

3

− q

2- y leexists



r

A

∗

2

− q

1- y leexists

β

2

1−β

2

− q



q q

β

1−β

− q

b

Figure13: Existen eofun lipped y les;

N = 4

.

Rememberthat

A = b + q

. Thus, if

q

is xed and

b

in reasesfrom

0

, un lipped y les haveorders

N

(see (11))and,possibly,

N + 1

,if

A

∗

N+1

− q > 0

. Later, as

b

in reases,the orderof y lesde reasesa ordingtoFig. 13.

Stability ofun lipped y les.

We intend to studythe mapping

ϕ

introdu ed just before Theorem 1. Sin e westudy

˜

ϕ

on entrateonadierentmapping:

Φ

k

_(v

0

) = β

k

(v

0

+ s

∗

+ 1)

dened for

v

0

∈ [βA, A]

underaxed

k ≥ 1

. Here

s

∗ △

_{= 0}

if

v

0

= A

;in ase

v

0

< A

,

s

∗

_{> 0}

istherstmomentwhen

y(s

∗

) = b

startingfrom

y(0) = b

,

v(0) = v

0

.

Lemma3







dΦ

k

_(v

0

)

dv

0





 < β

k

andhen e

Φ

k

isa ontra tion. Fun tion

Φ

k

isde reasing. Proof.Assumingthat

v

0

< A

,

s

∗

isasinglepositivesolutionto equation

(1 + A − v

0

)(1 − e

−s

) − s = 0,

(38) hen e

ds

∗

dv

0

=

1 − e

−s

∗

(1 + A − v

0

)e

−s

∗

− 1

=

1 − e

−s

∗

s

∗

1−e

−s∗

· e

−s

∗

− 1

and

dΦ

k

dv

0

= β

k

_{(1 +}

ds

∗

dv

0

) = β

k

_e

−s

∗

s

∗

− 1 + e

−s

∗

s

∗

_e

−s

∗

+ e

−s

∗

− 1

< 0.

Finally,

e

−s

∗

s

∗

− 1 + e

−s

∗

s

∗

_e

−s

∗

+ e

−s

∗

− 1

+ 1 = e

−s

∗

e

s

∗

− e

−s

∗

− 2s

∗

1 − e

−s

∗

− s

∗

_e

−s

∗

> 0,

be ausethenominatorin reases,startingfrom

0

at

s

∗

= 0

. Therefore

dΦ

k

dv

0

> (−β

k

₎

. Lemma4 (a)

A ∈



A

∗

K+1

,

β

K−1

1−β

K−1

i

i

d < βA

,where

d

isasolution to

Φ

K

_{(d) = A}

. Here andbelow,

β

K−1

1−β

K−1

△

= ∞

if

K = 1

;

K

isdenedby(31 ).

In this ase,

∀v

0

∈ [βA, A)

,the mapping

ϕ(v

˜

0

)

oin ides with

Φ

K

_(v

0

)

. (b) If

A ∈



β

K

1−β

K

, A

∗

K+1

i

thefollowing statementshold: (

α

)

∀v

0

∈ [βA, d]

,

ϕ(v

˜

0

) = Φ

K+1

_(v

0

) ∈ [βA, d]

; (

β

)

∀v

0

∈ (d, A)

,

ϕ(v

˜

0

) = Φ

K

_(v

0

) ∈ (d, A)

.

SeeFig.14. (Notethat,a ordingtothedenitionof

K

,

A ∈



β

K

1−β

K

,

β

K−1

1−β

K−1

i

;a ording toLemma 2 ,

A

∗

K+1

∈



β

K

1−β

K

,

β

K−1

1−β

K−1

i

.) Proof.(a)A ordingtothedenition,

d =

A

β

K

−s

∗

−1

,where

s

∗

solves(38)under

v

0

= d

. If

d = βA

then



(1 + A − βA)(1 − e

−s

∗

) =

s

∗

;

A

β

K

− s

∗

− 1 = βA,

(a)

-6

q

H

0

βA

A

z

βA

_V

A

v

0

2

z = v

0

z = ˜

ϕ(v

0

)

= Φ

K

_(v

0

)

(b)

-6

H

BB

@

q

q

0

βA

A

z

βA

_d

A

v

0

d

V

1

V

2

z = v

0

z = ˜

ϕ(v

0

) = Φ

K

(v

0

)

z = ˜

ϕ(v

0

) = Φ

K+1

(v

0

)

Figure14: Graphsof

ϕ(v

˜

0

)

. orequivalently

(

A =

β

_1−β

K

(s

∗

K+1

+1)

;

(1 + A −

β

K+1

1−β

(s

K+1

∗

+1)

)(1 − e

−s

∗

) =

s

∗

.

Toputitdierently,wehave

A = A

∗

K+1

if

d = βA

. Itremainstoprovethat

d − βA =

A

β

K

− s

∗

− 1 − βA

isade reasingfun tion of

A

. Sin e

s

∗

satisesequation



1 + A −

_β

A

K

+ s

∗

+ 1



(1 − e

−s

∗

) − s

∗

= 0,

ds

∗

dA

=

(1 − β

K

_{)(1 − e}

−s

∗

)

2

β

K

_e

−s

∗

_(e

_−s

∗

_{+ s}

∗

_{− 1)}

and

d(d − βA)

dA

=

1

β

K

−

ds

∗

dA

−β =

s

∗

e

−s

∗

+ e

−s

∗

− 1 + β

K

_{(1 − e}

−s

∗

)

2

_{− β}

K+1

_e

−s

∗

(e

−s

∗

+ s

∗

− 1)

β

K

_e

−s

∗

_(e

_−s

∗

_{+ s}

∗

_{− 1)}

.

Thedenominatorisobviouslypositivefor

s

∗

> 0

. Thenominatorequalszerowhen

s

∗

= 0

, itsderivativeequals